Synonyms: IV4 assumption, fourth MR assumption

This is one of a set of additional (to the core) IV assumptions that are required for a well-defined causal parameter.

Two types of monotonicity may be inferred to make causal inference in IV (including MR studies) - Deterministic monotonicity: often referred to simply as “monotonicity”, which assumes a monotonic relationship between the IV and exposure. In other words, a genetic IV could not increase the exposure in some people and decrease it in others. If this monotonicity holds, then the IV estimate is consistent with the ACE among compliers, where compliers are the subgroup of the sample affected by the genetic IV. Defining who this subgroup might be is unclear. - Stochastic monotonicity: this is a relaxation of deterministic monotonicity, because this only requires that a monotonic increasing association between the IV and the exposure exists conditionally on a set of covariates (which may or may not be measured). If this holds, then the IV estimate is consistent with a weighted average of treatment effects, such that more weight is given to treatment effects among subgroups where the effect of the IV on the exposure is greater. This assumption applies to both binary and continuous exposures but allows identifying an estimate that is less useful than the ACE among compliers. It is difficult in practice to know how important violation of these additional IV assumptions are in MR studies. Large GWAS collaborations increasingly combine results from many studies (though mostly from European original populations) and show consistency of association across these studies for variants defined as being associated with the exposure at genome-wide significance (variants used in most MR studies), suggesting homogeneity may exist for several MR studies in European populations. Non-parametric methods that provide bounds of causal effect estimates requiring only the core IV assumptions may be applicable for some MR studies. Triangulation of results with other (non-MR) methods is likely to improve causal inference from MR studies.

## References

- Sheehan NA, Didelez V. Epidemiology, genetic epidemiology and Mendelian randomisation: more need than ever to attend to detail. Human Genetics 2019.
- Swanson SA, Hernán MA. The challenging interpretation of instrumental variable estimates under monotonicity. International Journal of Epidemiology 2017;47:1289-1297.
- Small DS, Tan Z, Ramashai RR, Lorch SA, Brookhart MA. Instrumental Variable Estimation with a Stochastic Monotonicity Assumption. Statistical Science 2017;32:561-579.

## Other terms in 'Sources of bias and limitations in MR':

- Assortative mating
- Canalization
- Collider
- Collider bias
- Confounding
- Dynastic effects
- Exclusion restriction assumption
- Harmonization failure (in two-sample MR)
- Homogeneity Assumption
- Horizontal Pleiotropy
- Independence assumption
- InSIDE assumption (in two-sample MR using aggregate data)
- MR for testing critical or sensitive periods
- MR for testing developmental origins
- No effect modification assumption (Additional IV assumption)
- Non-linear effects
- Non-overlapping samples (in two-sample MR)
- Overfitting
- Pleiotropy
- Population stratification
- Regression dilution bias (attenuation by errors)
- Relevance assumption
- Reverse causality
- Same underlying population (in two-sample MR)
- Statistical power/efficiency
- Vertical Pleiotropy
- Weak instrument bias
- Winner's curse